Selection of Optimal Threshold and Near-Optimal Interval Using Profit Function and ROC Curve: A Risk Management Application

Abstract

The ongoing financial crisis has had major adverse impact on the credit market. As the financial crisis progresses, the skyrocketing unemployment rate puts more and more customers in such a position that they cannot pay back their credit debts. The deteriorating economic environment and growing pressures for revenue generation have led creditors to re-assess their existing portfolios. The credit re-assessment is to accurately estimate customers' behavior and distill information for credit decisions that differentiate bad customers from good customers. Lending institutions often need a specific rule for defining an optimal cut-off value to maximize revenue and minimize risk. In this dissertation research, I consider a problem in the broad area of credit risk management: the selection of critical thresholds, which comprises of the "optimal cut-off point" and an interval containing cut-off points near the optimal cut-off point (a "near-optimal interval"). These critical thresholds can be used in practice to adjust credit lines, to close accounts involuntarily, to re-price, etc. Better credit re-assessment practices are essential for banks to prevent loan loss in the future and restore the flow of credit to entrepreneurs and individuals. The Profit Function is introduced to estimate the optimal cut-off and the near-optimal interval, which are used to manage the credit risk in the financial industry. The credit scores of the good population and bad population are assumed from two distributions, with the same or different dispersion parameters. In a homoscedastic Normal-Normal model, a closed-form solution of optimal cut-off and some properties of optimal cut-off are provided for three possible shapes of the Profit Functions. The same methodology can be generalized to other distributions in the exponential family, including the heteroscedastic Normal-Normal Profit Function and the Gamma-Gamma Profit Function. It is shown that a Profit Function is a comprehensive tool in the selection of critical thresholds, and its solution can be found using easily implemented computing algorithms. The estimation of near-optimal interval is developed in three possible shapes of the bi-distributional Profit Function. The optimal cut-off has a closed-form formula, and the estimation results of near-optimal intervals can be simplified to this closed-form formula when the tolerance level is zero. Two nonparametric methods are introduced to estimate critical thresholds if the latent risk score is not from some known distribution. One method uses the Kernel density estimation method to derive a tabulated table, which is used to estimate the values of critical thresholds. A ROC Graphical method is also developed to estimate critical thresholds. In the theoretical portion of the dissertation, we use Taylor Series and the Delta method to develop the asymptotic distribution of the non-constrained optimal cut-off. We also use the Kernel density estimator to derive the asymptotic variance of the Profit function.